Question

1. acing the first test (1.8): below are scores (%) of mrs. liaos students on their first test: 43 45 72 73 78 80 81 82 82.5 85 85.5 86 86 87.5 87.5 88 88 89 91 91 92 93 93 93.5 93.5 94.5 94.5 95 96 98

a) make a boxplot to display the data.
b) how did the students do on mrs. liaos first test? use the graph from part (a) to explain.

Explanation:

Step1: Find the five - number summary

First, order the data. There are $n = 30$ data points. The minimum is $43$. The maximum is $98$. To find the median (Q2), since $n = 30$ (even), $Q2=\frac{87.5 + 88}{2}=87.75$. For the lower half (first 15 data - points), the median (Q1) is the 8th value, so $Q1 = 82$. For the upper half (last 15 data - points), the median (Q3) is the 8th value of the upper half, so $Q3=93.5$.

Step2: Draw the box - plot

On the number line, mark the minimum ($43$), Q1 ($82$), Q2 ($87.75$), Q3 ($93.5$), and the maximum ($98$). Draw a box from Q1 to Q3 with a line inside at Q2. Draw whiskers from the box to the minimum and maximum.

Step3: Analyze the box - plot for part (b)

The box - plot shows that the median score is $87.75$. The lower 25% of scores are below $82$ and the upper 25% of scores are above $93.5$. The inter - quartile range (IQR) is $Q3 - Q1=93.5 - 82 = 11.5$. The data is somewhat symmetric around the median, but with a slight right - skew as the distance from Q2 to Q3 is larger than the distance from Q1 to Q2. Overall, the students' scores are spread from $43$ to $98$, with a middle 50% of scores in the range $82$ to $93.5$.

Answer:

a) A box - plot is drawn with minimum at $43$, Q1 at $82$, Q2 at $87.75$, Q3 at $93.5$, and maximum at $98$.
b) The median score is $87.75$. The lower 25% of scores are below $82$ and the upper 25% are above $93.5$. The data has a slight right - skew and the middle 50% of scores are in the range $82$ to $93.5$.